Calculating the residue of a complex function

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Homework Help Overview

The discussion revolves around calculating the residue of a complex function at a specific pole, focusing on the function f(z) = (1 + z^2)^-3 and its pole at z = i. Participants are exploring the nature of the pole and the application of the residue theorem and Laurent series expansion.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the order of the pole, with one suggesting it is of order 3. There are attempts to apply the residue theorem and calculate the residue using limits, but some express difficulty in implementing the formula and understanding the process. Factorization is noted as a necessary step.

Discussion Status

The discussion is active, with participants providing hints and guidance on how to approach the calculation of the residue. There is an acknowledgment of the need to verify the order of the pole through limit calculations, and some participants are refining their understanding of the necessary steps.

Contextual Notes

Participants are working within the constraints of homework rules, which may limit the extent of direct assistance provided. There is an emphasis on understanding the underlying concepts rather than simply obtaining the answer.

AlBell
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Homework Statement



calculate the residue of the pole at z=i of the function

f(z)=(1+z^2)^-3

State the order of the pole

Homework Equations



I know the residue theorem and also the laurent series expansion but I'm having trouble applying these

The Attempt at a Solution



I think the pole is order 3 but I'm having trouble either expanding the function in order to find the a(-1) term or implementing the residue theorem and would appreciate some help! Thanks
 
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Do you know that the residue of a function f at the point a can be calculated by

\lim_{z\rightarrow a} (z-a)f(z)
 
I thought it was
dxzktg.png
?
 
AlBell said:
I thought it was
dxzktg.png
?

Yes, of course, sorry. My formula is for a simple pole.
So, can you use your formula to find the residue?
 
Yes but I am having trouble actually implementing the formula and wondered if anyone could do an example to help me understand it?
 
Well, first you'll need to determine the order of the pole.

Your guess is that the order of the pole is 3. So you'll need to look at

\lim_{z\rightarrow a} (z-a)^3f(z).

This limit should be nonzero for the order of the pole to be 3. If it is zero, then the pole is of a lower order. If the limit doesn't exist, then the pole is of higher order.

So, can you calculate that limit?

Hint: factor the denominator.
 
Ah I forgot about factorising, that makes it so much simpler! I get 3/8 I think!
 
VERY close. You probably got something like 12*(2i)^(-5), right? Just compute this again. The 3/8 is right, but you are missing two factors.
 

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